On Two-Stage Householder Orthogonalization

TL;DR

This paper introduces a two-stage Householder orthogonalization method that is unconditionally stable and efficient for orthogonalizing matrices against a set of orthonormal columns, improving numerical stability in Krylov subspace algorithms.

Contribution

The paper proposes a novel two-stage Householder orthogonalization algorithm that leverages the generalized Householder transformation for improved stability and efficiency.

Findings

The method is unconditionally stable based on theoretical analysis.

Numerical experiments confirm the stability and efficiency of the proposed algorithm.

Only a square submatrix of V needs to be orthogonalized, reducing computational complexity.

Abstract

Two-stage orthogonalization is essential in numerical algorithms such as Krylov subspace methods. For this task we need to orthogonalize a matrix $A$ against another matrix $V$ with orthonormal columns. A common approach is to employ the block Gram--Schmidt algorithm. However, its stability largely depends on the condition number of $[V,A]$. While performing a Householder orthogonalization on $[V,A]$ is unconditionally stable, it does not utilize the knowledge that $V$ has orthonormal columns. To address these issues, we propose a two-stage Householder orthogonalization algorithm based on the generalized Householder transformation. Instead of explicitly orthogonalizing the entire $V$, our algorithm only needs to orthogonalizes a square submatrix of $V$. Theoretical analysis and numerical experiments demonstrate that our method is also unconditionally stable.